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THE JOURNAL OF FINANCE ¢ VOL. XLVIII, NO. 3 « JULY 1993

Measuring Asset Values for CashSettlement in Derivative Markets:

Hedonic Repeated MeasuresIndicesand Perpetual Futures

ROBERT J. SHILLER*

ABSTRACT

Two proposals are made that may facilitate the creation of derivative marketinstruments, such as futures contracts, cash settled based on economic indices. The

first proposal concerns index number construction: indices based on infrequentmeasurements of nonstandardized items may control for quality change by using a

hedonic repeated measures method, an index number construction method thatfollows individual assets or subjects through time and also takes account of mea-

sured quality variables. The second proposal is to establish markets for perpetual

claims on cash flows matching indices of dividends or rents. Such markets mayhelp

us to measure the prices of the assets generating these dividends or rents even

when the underlying asset prices are difficult or impossible to observe directly. Aperpetual futures contract is proposed that would cash settle every day in terms of

both the changein the futures price and the dividend or rent index for that day.

THIS PAPER CONCERNS PROBLEMS in the measurement of asset values orpresent values, problems that, if solved, may facilitate the creation of newhedging vehicles, new cash-settled futures, options, and other derivativefinancial products.The greatest components of world wealth are not hedgeable at all, and

measurement problems may be an important reason why they are not.Human capital is by far the greatest component of wealth, properly mea-sured, and yet there are no liquid markets on which humancapital risk canbe hedged. Real estate represents most of the wealth as conventionallymeasured, and yet here there are no liquid derivative markets (except onmortgages).! Privately held financial assets are also of great importance,especially outside the United States, yet there are no derivative marketsspecifically designed for hedging the risks of this class of investments. There

would logically be derivative markets representing ‘the major components of

* Cowles Foundation, Yale University. The author wishes to thank Peter Abken, Steven Bloom,

Michael Brennan, John Campbell, P. H. Kevin Chang, Karl Case, Zvi Griliches, Sanford

Grossman, Jonathan Ingersoll, Paul Kupiec, Hayne Leland, Charles Longfield, Ben Krause,

Nathan Most, Stephen Ross, Jeremy Siegel, Christopher Sims, Steven Sural, and Allan Weiss for

helpful discussions. Jose Carvalho, Ivailo Izvorski, and Toshiaki Watanabe provided research

assistance. This research was supported by the National Science Foundation under grant No.

SES-9122229.

! See Case, Shiller, and Weiss (19993).

911

912 The Journal ofFinance

the producer price index or consumerprice index, but none exists. A futuresmarket in the aggregate consumerprice index has succeeded only in Brazil, a

country where the volatility of the consumer price index has been very high.There are many other economic indicators that represent specific risks facedby economic agents, risks for which there is no hedging market today.There are two separate (but related) measurement problems that may be

an important part of the reason whythese cash-settled derivative marketshave not been established and successful.The first problem is that the underlying cash marketprices, in responseto

which a derivative market would cash settle, may be observed only infre-quently and, when they are observed, the observations at different times areon assets of different type of quality, so that their average price does notallow us to infer the changes through time in the price of an existing asset.This problem may account for the slowness to develop derivative markets on

real estate prices or prices of privately held financial assets. These assetstend to be held for years or even decades between sales, making the nontrad-

ing problem of near-astronomical proportions. There are secular changes inthe stock of outstanding real estate, such as a gradual increase in the size orquality of properties.The second problem is that in manycases the best (or only) measurements

we have on the cash market are not on asset prices at all, but are better

thought of as measurements of the dividend or rent on the asset. Thisproblem may be an important part of the reason for the absence of futuresmarkets in such things as labor costs, commercial real estate, the consumer

price index, or components of the producerprice index. The measurements wehave on these are more on cash flows than on asset values or present values.No prices exist at all for the asset value of labor costs: people are not boughtand sold. Now, of course, we could easily construct a conventional] futures

market to be settled on a wage index, but that would be a market settled onthe flow return on humancapital rather than its price.Most financial futures contracts today settle on the price of an enduring

asset, rather than on a dividend or rent paid on the asset. One might imaginethat the exchangescould easily have created conventional futures markets onthe dividend accruing to a stock price index as well as the index itself, butthere has apparently been no interest in such contracts. While we cannot ruleout that some people will want to trade conventional futures on the dividendaccruing to a portfolio of stocks, there is probably reason to suspect that most

would prefer to trade on the portfolio price itself. Present values collapseinformation about the indefinite future into today’s price.For the first problem, a hedonic repeated measures price index number

construction method is proposed here. There has been much discussion of theproblem of quality change in the literature on the construction of producerand consumerprice indices. But this literature does not solve the problem ofunobserved quality change for the purpose of constructing indices for thesettlement of contracts. Discussions of these indices presume that they aretrying to measure the price of the flow of newly produced commodities. The

Measuring Asset Value 913

problem of quality change is then fundamental and deep, since the qualitiesof the commodities change through time in unobserved or unquantifiableways; the indices are in effect pricing apples one period and orangesthe next.The hedonic repeated measuresprice index proposed here follows the price ofexisting assets through time taking account of their quality. Many of thecomponents of the producer price index (such as airplanes, railroad equip-ment, computers) could be priced in this way. This may produce indices thatare radically different; in some cases prices might fall dramatically throughtime even though the conventional producer price index component does not.The constructors of the producer price index may be resistant to changingtheir methods so radically, but for the purpose of constructing hedginginstruments we may need to do just this.The hedonic repeated measures index number construction method gener-

alizes the repeat sales price indices of Baily, Muth, and Nourse (1963), Case

and Shiller (1987, 1989), Webb (1988), and Goetzman (1990). The generaliza-tion here is to take account of hedonic variables (variables measuring thequality of each asset sold or time measured) within the context of a repeatedmeasures index number construction method. The repeat sales price indiceswere improvementsover earlier indices (simple averages or mediansofpricesof items sold) in that the repeat sales index number construction methodsfollowed individual properties through time, so changes in the index occurredonly in response to changesin prices of individual properties sold. The modelsthat give rise to these indices assumed that all kinds of properties had thesame price path (up to a scalar multiple and an idiosycratic error term)through time. Consider in contrast financial indices such as stock priceindices. The constructors of these financial indices recognize that differentstocks (items of different quality) may have different price paths throughtime, and seek to construct an index representing the price path of astandard portfolio of stocks. The method discussed here allows us to do thiswith assets where sales occur infrequently and where the measure of qualitymay not be categorical. It is a response to somecritics of repeat sales indices

(Abraham (1990), Case, Pollakowski, and Wachter (1992), and Clapp andGiaccotto (1992)) who expressed concern that those properties represented byrepeat sales may not always be representative of the market.The second proposal, for the problem that dividends or rents rather than

prices may be well measured,is to create markets that will provide measures

of asset prices: to create markets for perpetual claims on cash flows, paidfrom shorts to longs, representing the dividends or rents on assets. Specifi-cally, perpetual futures contracts are proposed here that would cash settle onindices of dividends accruing to an asset in such a way that the futures priceshould tend to track the value of the asset that generates the dividend. Eachcontract is perpetual so that it can price the entire stream of dividendsaccruing to an asset, and thereby provideprice discovery for that asset. Also,with perpetual contracts, any given contract does not grow shorter term withtime, and so a single contract can be adopted as a standard through time. Thefutures markets institutions of margin accounts and daily resettlement allow

y14 The Journal ofFinance

perpetual contracts to be traded even though no marketparticipant, short orlong, can assure credit worthiness in the distant future and each participanthas only a finite horizon interest in the market. Once a perpetual futuresmarket is established and liquid, the futures price may be used as a measureof asset value in other derivative markets, such as forwards or options.Past perpetual claims contracts that are analogous to perpetual futures

have traded only where the cash market is very liquid and cash prices areeasily observable. It is proposed here that some perpetual futures contracts,such as a market for human capital, could even be traded “blind,” the market

at first never even having observed the cash price. This may seem like aradical proposal, but, of course, every initial public offering of stock is firsttraded with little knowledge of the ultimate market price, and there is noconceptual difference here in the problems facing traders.

I. Hedonic Repeated Measures Index Number Construction

A. Repeated Measurement Design

Students of experimental design have long advocated repeated measuresmethods for their greater “design power” (see for example Lee (1975), andDunn and Clark (1987)).? For example, in testing whether a drug has aneffect on human subjects, it is often better to use information only on subjectswho were in both the experimental and control groups at different times,thereby ruling out the possibility that the results were confounded by unob-served differences between the contro] and experimental groups. This isespecially important when it is impossible to control membership in the two

groups completely (as when some subjects drop out of the study). Theimportance of repeated measures methods will be especially important in thefinancial applications intended here, as we may havelittle or no control overthe dates when prices, dividends, or rents are observed."Constructors of financial price indices also use repeated measures methods,

without calling them by this name. Theyfollow the individual stocks or bondsthrough time, linking them out or replacing them with new ones when theydisappear or become unsuitable for the index, thereby maintaining a re-peated measures design. No one would advocate computing a stock priceindex that was the average price of one sample of stocks for one period and ofa different sample of stocks the next period, even if some objective measures

of the firms (bock values, industry category, etc.) were the same acrossperiods. That is, however, essentially what is done for the most widely quotedprice indices for residential real estate. The median sale price of a singlefamily home published by the National Association of Realtors is based on

* For the same reason, researchers have long appreciated the importance of panel data for

empirical work (see for example Hsiao (1986)).

* The literature on hedonic regressions has shown some waysof using data onsales in the used

market to correct for quality changes (see Cagan (1971) and Hall (1971)).

Measuring Asset Value Y1ld

selling prices of whatever houses were sold each time period, and makes noeffort to standardize houses from period to period. The constant quality indexpublished by the Department of Commerceis based on a regression-per-periodhedonic regression involving a numberof characteristics, and the regressionmethod assures that in a sense, in terms of these measurable characteristics,standard houses are priced. But, especially since the constant quality indexprices only new houses, there is a distinct possibility for bias because ofunobserved quality changes. Consider a period when housing prices in anarea have dropped very far, so far as to be below construction costs. Therewill still be some construction, as houses are built for special purposes and inspecial areas, and the prices of these houses will not be representative ofallhouses; indeed they will certainly tend to sell for prices at or above theconstruction costs. These houses may not differ from most houses in terms ofobjective measurements like the numberof rooms.*

B. Hedonic Repeated Measures Indices

It is instructive first to review the conventional hedonic methods. Theconventional regression-per-period hedonic index is produced by regressing,each time period, log price (or log dividend or log rent) on a constant and anumber of variables, called hedonic or quality variables, that characterize the

property sold or rented at the time of observations.” The dependent variablefor time ¢, call here Y,, is an N,-element column vector, where N, is thenumber of observations of prices or rents at time ¢, and where the ithelement of Y, is the ith log price or (or log dividend or log rent). The matrix ofindependent variables Z, is an N, X K matrix whose ith row consists of (aconstant and) a vector of hedonic variables for the time of observation of thatsale or rent. The regression model is Y, = Z,y, + €,, where y, is a K-elementvector of regression coefficients and e, is an N,-element vector of error termswith mean zero and variance matrix 0,. The generalized least squaresestimate of y, is ¥, = (Z{07'Z,)'Zi07'Y,, which simplifies in the casewhere ©, is proportional to the identity matrix to ordinary least squares4, =(Z)Z,)*ZY,. A regression-per-period index I.» for time ¢ may be takenas a fitted value of the regression for some standard property whose charac-teristics are given by the 1 x K element vector Z: Leppt = Zy,. Or, chainindices could be constructed from the regression coefficients, where thequality of the standard property is updated through time.For example, in the context of real estate prices, if p,, is the price of

property z observed sold at time ¢, if N, = 3, and Z, consists of a constant

* The problem of unmeasured quality changes is an important reason why hedonic methods

are not used more widely (see Triplett (1990)).> Using the log price as a dependent variable creates a log price index whose antilog is

essentially a geometric average of prices or rents. For financial market applications, it may be

preferable to use arithmetic indices; arithmetic analogues of all the indices described here are

possible (see Shiller (1991, 1993)).

916 The JournalofFinance

and the numberof square feet for that property at time ¢, then the matriceare:

Pu 1 8s,

Y, = Pas Z,= {1 8p, (1)Pst 1 83

where s;, is the square feet of floor space in property i at time tf.It is convenient, for purposes that will become clear in a moment, to

assemble these regression matrices for 7’ time periods into one giant regres-sion to be run wherethe price indices are computed for all 7 times for whichwe will have the index. We construct the N-element vector Y, where N = 3.N,;Y is the stacked Y, vectors. We also construct the N x TK matrix Z whichisblock diagonal, with the Z, matrices along the diagonal. Now, the combined

regressions can be written as a single regression model Y = Zy + € where yis the stacked y, vectors, so that y has TK elements, and e is an N-elementvector of error terms with mean zero and variance matrix 1). If 0 is blockdiagonal, then since the Z matrix is block diagonal, the estimated coefficientvector ¥ = (Z'0Q7'Z)"'Z'O-'Y is just the stacked per period regressioncoefficient vectors.For example suppose that there are only T = three timeperiods,times 0, 1,

and 2, and the matricesare:

Y Z, 0 0Y=|¥Y,| Zz=|0 Z, 0 (2)

Y, 0 0 &Z,

As time goes on, we would keep expanding the Y and Z matrices byappending the latest Y, to Y and the latest Z, to the bottom right corner of amatrix whose rows and columns have been augmented from Z with a rowand column of zero matrices: of course, so long as © is block diagonal, thisentails no revisions in past values of the index computed before.The fundamental problem with hedonic price indices such as theseis, as

noted in the introduction, that we do not have all characteristics as hedonicvariables; there are likely to be omitted hedonic variables. Thus, we seek arepeated measures method.The repeated measures approach that we shall pursue here consists merely

of adding additional dummy variables, variables that proxy for the omittedhedonic variables, to the giant regression. We will, following the literature onexperimental design, call them subject dummies, even though they mightbetter be called property dummies or asset dummies in many ofour intendedapplications. There will be one dummyfor each subject: one for each individ-ual property (if we are estimating a real estate price index or, say, the

railroad equipment component of the producerprice index), individual person


(if we are estimating a labor cost index), or individual plot of land (if we areestimating a yield per acre index), in each case a dummythat identifies thatsubject. Each dummyis zero everywhere except for an observation where thedependent variable element corresponds to that property or individual; therethe dummy is 1.00. These subject dummy variables are indicators of theunique quality of each property or individual. To avoid multicollinearity, wewill, when adding the subject dummies, drop the first column of Z,, so thatwe will have an augmented matrix Z, whichis N x TK — 1 + m where m isthe numberof subjects. The regession model is now Y = Z,B + € where B isa TK — 1+ m elementvectorof coefficients, and ¢ is, as before, an N-element

vector of error terms with mean 0 and variance matrix 0; the generalizedleast squares estimate of B is B = (Z',0°'Z,)"'Z,.Q7'Y.For example, we may augment the Z matrix in (2) to produce the aug-

mented matrix Z,:

[2 0 0 Dy Do Dno|Za = 0 Z, 0 Di, Dy, vt. D1 (3)

0 0 4, Dy Dy Dno

Where D,, is the N, X 1 subject dummy for subject k at time ¢. For example,with the matrices (1) above D,, equals [0,1,0]' and D,, equals [0, 0, 0)’. Since

the sum over k of the subject dummies equals the vector 1, there would bemulticollinearity in this regression if we included all Z, matrices completely,so we will suppose that the constant term is dropped from Z, (hence the*above the Z, in (3)). Adding the subject dummies as columnsto produce theZ, matrix above will generally break the block diagonality of the matrix, andunless the subject dummy variables have a certain conformation, will cause

the index number production method to produce revisions in lagged values ofthe index.®Note the analogy of this regression model to the conventional fixed effects

analysis of covariance model (or, in the special case here where there are noquantitative hedonic variables, only constant terms in the Z,, the analysis ofvariance model). By this interpretation, the subject dummies correspond tothe “experimental factors” and the hedonic variables to the “concomitantfactors.” The fixed effect formulation embodied in the subject dummies,

rather than the random effect formulation, is adopted here to allow for thepossibility that the mix of properties or items changes through time. Aconventional random effect model, for which we would substitute in place ofthe subject dummies a variance component to the error term related to the

° ven if we did not include subject dummies, we would then properly need to take account of

subject variance components in 9, which would cause revisions even in regression-per-periodmethods. Ways of handling revisions in data from the standpoint of contract settlement are

discussed in Shiller (1993).

Y1S ihe Journal of Finance

property, would normally assumethat the distribution of the property-specificnoise did not change throughtime.

This N x (TK —1+m) matrix Z, may be very large, both in terms ofnumbers of rows and numbers of columns. Particularly problematic is thatthere is a column for every individual property or item and then some; thismeans that the Z,Q~'Z, matrix may be of very large rank, and thus veryhard to invert. When estimating residential real estate price indexes, forexample, we may have data on millions of houses, resulting in a Z',07'Z,matrix whose rankis in the millions. Fortunately, however, since our interestis in the coefficients of the original Z, variables and not in the coefficients ofthe dummyvariables, we can exploit the structure of the subject dummiesinsuch a way as to reducethe size of the matrix that must be inverted.To do this, construct a matrix S, such that y =S,Y is the vector of

differences of consecutive observations of the dependent variable for eachsubject. For example, following the real estate example above, if individualproperty i appears three times, in times t,, t), and t,, then there will be two

weesrows in Y= SY correspondingtLO lL, a rowwith element Pia ~ Piro and a row

with element p;;. — P;.;. Now S, is of dimension n x N where n is thenumberof pairs of consecutive observations of individual subjects that can beconstructed out of these N observations. Now construct an m X N matrix S,such that Y, =S,Y is the vector of all first oseof individualsubjects. Note that N =m +n and that the matrix S = (S), S)), is nonsin-gular. Call Y= SY, Z =SZ,, and 0 = SQS’. Let us denote the upperleftn X (TK — 1) corner of Z by z, the upper (TK — 1) — element vector of B byB, and the upperleft n x n block of 0 by 0,,.

It will be shown now that the generalized least squares estimate(z'O7)2)12'O;)y is the same as first component $B of B =ZO,“1a7.0“1Y. The former expression involves matrices of smallerdimension than the latter.To show this, note that Z = SZ, is block triangular. The upper right

n X m block of Z consists of zeros. The lower right m X m block of Z is theidentity matrix. The generalized least squares estimator B =(Z/,07'Z,)°1Z,.0-1Y equals (Z'SZ)-1Z'SY where § = O-! = 8" 'Q71S7};partition > into 4 square parts, with upper left n x n part called %,,,etc.From the last m normal equations for B (the last m rows of Z'SZB=Z'Y)one finds that B, — Y, + Z,, B = 3322.,(y — zB); substituting this into thefirst TK —1 normal equations one finds that 2'(24, — SSeS.)28 =2'(X4, — Sy. Eo! Zo1)y, and so B = (z’/Qj'z)~!z2’Qy,|y as was to be shown.’

For an example of this hedonic repeated measures regression, supposethatthe matrices Z, contain two columns, a vector of ones and the variables,,(square feet of floor space of ith property at time ¢) as in (1), and let us

; Incidentially, had we begun with a random effect rather than fixed effect formulation in (3),the single-sale properties would not have dropped out of the estimation as they have here,

reflecting the implicit assumption of the random effect model that the mix of properties does notchange through time,


premultiply the regression independent variables and dependent variable bythe matrix S described above. We may then be left with the regression (for asample of six houses) whichis:

1 0 Sig $11 0 Pir ~ Pio

1 0 S99 891 0 Pai — Poo

_ -1l oil 0 ~ 831 S32 y= P32 — P31 (4)

~-1l oil 0 —~S41 $4 Pao — Par

0 1 ~ 85 0 559 P52 — Pso0

0 1 ~S¢p 0 S62 Pea ~ Peo

The hedonic repeated measures regression model y = zB + € (where e is ann-element vector of error terms with mean zero and variance matrix 1,,) hasa simple interpretation.® In (4), the change in log price between any twoperiods depends on the time periods of the two observations, and on thesizeof the house at the two times; it may depend on thesize of the house eitherbecausedifferent size houses have different price paths or becausethe size ofthis house changed between sales. The hedonic repeated measures index willbe constructed using the coefficient vector B = (2'O7)z)~1z'O7,'y. The hedo-nic repeated measures index at time ¢ could then be taken, in this example,as the fitted value for a standard size (square feet = 5) house at time ft: forexample, a fixed weight index at time ¢ would be the coefficient of theconstant term for time f (or zero if t = 0) plus § timesthecoefficient of thesquare foot variable for time ¢. (The index may be adjusted by an additiveconstant to make it equal an assigned value in the base period.) Other kindsof hedonic repeated measures indices could be constructed from the estimatedregession coefficient 8: chain indices or indices representing the value of aportfolio reinvested through time to be representative of the market in termsof the hedonic variables.Note that even though we have not used any information about the

characteristics of individual properties per se, our set of subject dummyvariables spans the set of any characteristics that are constant for eachindividual asset through time and that have a constant proportional effect onprice or rent. That is, continuing our real estate example, suppose that at a

later date, after the regression was run, someone discovers that a very

important hedonic variable was left out of the regression, a variable that

° The transformation that produced (4) is analogous to the differencing employed in many

panel data analyses (see for example Hsiao (1986)), but here the differencing interval may bedetermined by random dates of sales and may not be constant across properties or subjects. Note

that if we dropped the last three columns of z, corresponding to the hedonic variable, then this

regression reduces to that of Baily, Muth, and Nourse (1963). The method using(4) for regession

differs from the hedonic repeat sales regression method proposed by Case and Quigley (1991) inthat (4) does not make coefficients linear in time, and in that (4) makes no useof single-sales

data. Problems in bias due to errors and variables may be magnified by the differencing here; see’Griliches and Hausman (1983),


measures the quality of a house independently of the number of square feet,and it was found that at some times higher quality houses are sold than atother times. Because of concern that the price index might jump up erro-neously in periods when the mix ofsales is relatively tilted towards the highquality houses, we consider adding a single column to the Z, matrix in (3)that represents this quality variable for each property. But, if we tried doingthat, we would quickly discover that we would have to take that additionalcolumn out of the Z, matrix. So long as quality is constant through time foreach house, there is always a weighted sum of the subject dummy columnsofthe Z, matrix (3) that will equal the particular hedonic variable. This meansthat the repeated measures regression had already in effect. taken this newhedonic variable into account. In fact, so long as we can assume that thecoefficient of the hedonic variable is constant through time, we have in thematrices (3) accounted for all such hedonic variables, all nonlinearities in

these variables, and all interaction effects between them; there is no arbi-trariness in our methods. No one can ever claim to have found anothercharacteristic variable or interaction effect that was left out of this regres-sion. However, the additional hedonic variable maystill be of use for us ingauging whetherthe relative price of the higher quality homes varies throughtime: if we add not a single column to the Z, matrix in (3) for this variable,but a single column to each of the Z, matrices exemplified by (1), thengenerally none of the columnsthat are created in (3) will be collinear with thesubject dummy variables.

If any characteristic is constant through timefor all subjects (houses), thenthe sum of the hedonic variable columns in z corresponding to these charac-teristics (e.g., the sum of the last three columns in z in (4)) is zero, and so theregression cannot be run due to multicollinearity. This poses no particularproblems however; if all characteristics are constant through time for eachsubject, we need only drop one of these columns. There is no reason to dropall of them. Keeping the others in this example allows us to correct forpossible changes throughtime in the pricing of individual characteristics. Forexample, in constructing a real estate price index, there may be concern that

the price path of big houses is different from that of little houses, and thatthere are times when there are many big houses sold and times when few bighouses are sold. An index number based on thefitted value of this regressioncould take account of a standard house in terms of square feet. Alternatively,there could be in each single period hedonic regression a “big house” dummyvariable in place of the square foot variable, or there could be dummies forvarious types of houses.’ (This would make our method essentially that ofcomputing an index for each type of house, so that a fixed weight aggregateindex can be computed from these indices, just as aggregate stock price

* Clapp and Giacotto (1992) have stressed that assessed value might be used in a repeat salesreal estate price index where assessed value in effect replaces one of the sale dates: it may be

even more useful to use assessed value in a given year in place of square feet in (4) as a hedonic

variable.


indexes are produced from prices of individual stocks using weights represen-tative of quantities outstanding.) All these methods will allow us to constructindices that control for time variation in the price of characteristics.

Applying this method as exemplified in (4) might produce very differentindex values than do conventional index number construction methods. Theprice or rent index produced may tend to decline through time due todepreciation of the individual assets. This decline in the index is in no way aproblem for an index whose purposeis to serve as the basis of trade for thesettlement of contracts. In fact, having depreciation included as part of theindex could be considered an advantage, as it could allow hedgers to dealwith the intrinsic uncertainty about depreciation. We might also want toinclude as a hedonic variable the age of the asset, and use an expressionlike(4) to produce an index of prices of or rents on an asset of standard age,although this would not allow us to correct for any overall downward bias inthe index due to depreciation. One would have to drop one of the columnsinthe z matrix corresponding to the age of the asset, since the sum of thesecolumns would equal the sum (for the columnsof z corresponding to constantterms, the first two columnsin (4)) of i times the ith column of z. One cannotgauge the absolute effect of asset age on price or rent in a hedonic repeatedmeasures regression, since all assets age the same amount between the sameintervals of time; there is no way to distinguish the effects of age from theeffects of price or rent change of a standard age.” But it is not necessary todrop all of the columns in the z matrix corresponding to age variables:leaving all but one of them in allows us to account for the possibility thatassets of different ages have different price paths through time, which couldaffect the index if the age mix of assets sold changes through time.

C. Applications of Hedonic Repeated Measures

The simple framework exemplified by (4) appears not to be in the indexnumberliterature; its use might have had many applications. It is immedi-ately applicable, for example, to real estate price indices, where there arelong sales histories of individual properties available in deeds offices, It couldbe used to produce an index of prices of infrequently traded bonds; thehedonic variable could be time to maturity, so that the hedonic repeat salesmethod could use data on adjacent maturities to price a bond of a standardmaturity. It would no longer be necessary to base indices only on that subsetof bonds that are frequently traded in a fixed maturity category.

The framework would also be helpful in constructing labor cost indices. TheBureau of Labor Statistics Employment Cost Indices are computed accordingto a standard Laspeyres index formula from data on wages in occupational

categories (see Sheifer (1975) and Wood (1982)). No attempt is made to adjustfor quality characteristics of the workers in the occupational categories, or to

© This pot was made by Bailey, Muth, and Nourse (1963) and Palmquist (1979); see also

Hall (1971).


base the index on changes in individuals’ wages. Blanchard and Katz (1992),who sought to develop, from Current Population Survey data on individuals,a wage index that controls for industry and worker characteristics, regressedlog hourly wages on linear, quadratic, cubic, and quartic experience terms,dummyvariables capturing an individual’s education level, race, urban-ruralresidence, and full- or part-time work status, as well as the occupation andindustry in which the individual is employed. Although their method showssome Improvements over the Bureau of Labor Statistics procedures, they didnot use a repeated measures formulation either. When data are coliected witha rotation method, sampling at intervals from the same households as with

the Current Population Survey, hedonic repeated measures indices would bepossible.

II. Perpetual Futures

A. Definition and Settlement Procedures

To create a market for the present value of a cash flow represented by somedividend or rent index, we need to create a perpetual claim on a cash flowrepresented by the index. A daily dividend index, ideally based on dividendsactually paid each day, should be constructed for this purpose, even if thereare many zeros in the dividendseries, and dividend payments are measuredas occurring in bigger, infrequent, lumps. On dates where the daily dividendindex is nonzero, shorts in the perpetual claims market pay a quantity ofmoney proportional to the index to longs.

One kind of perpetual claim will be called here perpetual futures. Withperpetual futures, on any day ¢, the next daily resettlement s,,, received bya long from the short in the perpetual futures contract is defined to be:

Siri = (har -f + (dy - if) (5)

where f, is the perpetual futures price at day ¢, d,,., 1s the index, represent-ing dividends actually paid to owners of the underlying asset on day ¢ + 1,and r, is a return on an alternative asset between time / and time ¢ + 1. Thealternative asset would be any liquid asset, though in practice it would mostlikely be the return on risk-free short debt, such as an overnight repo rate,

debt that is either nominal or indexed to a consumerprice index.Wewill generally expect that d, and r, are both small relative to the price

of the asset on which the dividends are paid. The dividend d, consists only ofdividends actually paid on date d,, a small amount in general. The alterna-tive asset return r, is a daily interest rate (something like, say, 0.0005).Thus, the daily resettlement wili likely be primarily just the change infutures price since the preceding day. Still, the component of the dailyresettlement due to the difference between dividend d,,, and r,f, is offundamental importance, and in fact it is this component that ultimately

determines the futures price.


The terms on the right hand side of the expression are grouped so that thefirst term (f,,, — f,) corresponds to the daily resettlement of a conventionalfutures contract and the second term (d,,, — r,f,) corresponds to the finalcash settlement of a conventional futures contract. In a conventional cash-set-

tled futures contract with a fixed maturity, there is every day except

for the last day a daily resettlement f,,, —f, based only on the futuresprice; this daily resettlement is replaced by a final cash settlement cp — fyp_,on the maturity date IT, where C, is the final cash price. In a perpetualfutures contract, both types of settlement occur every day. In a perpetual

futures contract, the term corresponding to the final cash settlement is notd,., —f, but d,,, minus something akin to a “permanent” dividend on anasset with price f,, inferred by multiplying f, by r;.

There is also a different interpretation of expression (5). Suppose we took f,to be the price of an asset that is a perpetual claim, iie., a contract,"

promising the buyer of the contract a perpetual stream of dividends (paid by

the writer of the perpetual claim) equal to the index d,,,% = 1,2,... Long-term indexed government bonds, such as the index-linked gilts in the UnitedKingdom which are linked to the U.K. Retail Price Index, are, although theyare finite term, essentially such contracts between the government and thepublic. Then s,,, would just be the excess return to a one-period investmentin one unit of the perpetual claim, the one-period return to a portfolio that islong one perpetual claim and short an equivalent value of the competingasset. Expression (5) is the return a long would obtain if he or she bought theperpetual claim and borrowed the entire purchase price on margin, andminus the return a short would obtain if he or she sold short the perpetualclaim with 100 percent margin, where the rate earned on margin accounts isr,. In practice, longs could not borrow 100 percent on margin and uncoveredshorts would be required to hold more than 100 percent margin.” I haveemphasized here the kind of perpetual claims called here perpetual futuresbecause of the expectation that the contract might best be handled in afutures market without any short sales of securities.

B. Relation between Cash and Futures Price

Since the perpetual futures contract is essentially a perpetual claim pur-chased with a margin loan, a long can undo this loan by investing an amount

f, in the alternative asset at the time of purchaseof the futures contract, anda short can undo his or herside of the loan by shorting the alternative asset.If this is done every period forever, then, so long as the futures price f, does

"! This is not a forward contract, not a contract to pay f, in the future; the distinction between

forwards and futures stressed by Cox, Ingersoll, and Ross (1981) does not arise here.

2 Regulation 7 provides that 150 percent margin is required for uncovered short sales ofsecurities; the regulation of course does not specify what would constitute adequate cover for onewhois short a perpetual claim on an economic index. For someof the contracts considered here

such cover might be hardto define.


not diverge in such a way that the present value at ¢ of f,,, does not tend tozero as k goes to infinity, the long may be thought of as receiving (or theshort paying) the same dividend payments as the holder of the asset thatpays d,. This suggests that, if there is an observed price p, on a liquid assetthat pays the dividend stream d,,,k = 1,...,then the futures price f, willtend to equal the price p,.”°Note that the argument that the futures price f, will tend to equal the cash

price p, does not hinge on what alternative asset, whose return r, is part ofthe basis for cash settlement, is chosen by the exchange for the contractspecification, so long as r, is the return on a marketable, liquid asset.However, the choice of the alternative asset is not irrelevant, as it affects the

cash settlement. A perpetual futures contract is analogous to a swap of thedividend stream d,,,, 8 = 0,... with the return of another asset, and whiletraders can undo the swap by other portfolio transactions, the lowest costthing to do is to just buy or sell the futures contract. For this reason, theusual alternative asset would probably be a repo rate, or other nearly risklessobligation. If riskless indexed short debt is available. then it might bOO1Galion. if TISKiGSS INGexeG SNOTtl Geo. 1S avaluagie, NON It might pe

preferable to use this as an alternative asset, since then hedgers would behedging the real, rather than nominal, value of the asset.This argument that f, will tend to equal p, does not quite have the same

force as a no-arbitrage condition. At time f, anticipating any finite holdingperiod of length k, one always faces the risk that the terminal futures pricef,., Will not equal the cash price p,,,. This suggests that f, might track p,(supposing that the latter is observed) less well than do other futures pricetrack their cash prices.

C. Liquidity and Long Investing Horizons

The perpetual futures contract is designed so that there can be liquidity ina single long-term contract. There would perhaps be no other contracts tradedfor this asset, no shorter horizon futures contracts, thereby forcing thevolumeof trade to concentrate on a contract with a long horizon. In contrast,there is a risk that a conventional futures market involving long horizonmaturities will wind up having most of the volumeof trade and open interestin the short maturities, even if hedgers would, other things equal, prefer tohave long horizon contracts. The long horizon futures contract might fail toobtain liquidity, and thus not serve the needs of hedgers. Perpetual futurescontracts may tend to attract liquidity, even if there are shorter horizoneontrarta aimnn at represents a fixed referen ce noint an whir 1WLU acts, since 10 4 wht AahOkeBEI a bee) Weehw pvr AJELE ¥¥ ALLWAL LLY

attention might be directed."

8 Another kind of perpetual futures contract might involve announcing r, at something other

than a market return; if r, were set above a market interest rate, this might have the effect of

shortening the effective maturity of the perpetual futures contract.Essentially the same advantage was claimed for undated futures by Gehr (1988) and

perpetual currency options by Garman (1987).


The futures exchanges have in effect given away the business of long-datedcontracts to the over-the-counter markets. This may be because the futuresexchanges do not have any meansto insure that in market equilibrium it willbe the long-term market that is selected to have the high volumeoftrade.Their policy of scheduling contract months at frequent intervals meansthatthere is no natural single long-term contact on which to base trading; thustrading gravitates to the shortest term contract. It is natural for long horizoncustomers to do business in the over-the-counter markets: for them there isno liquidity advantage in the long horizon conventional futures markets tooffset the extra services and tailoring of the contract to the customer’s needsfound in the over-the-counter markets.The clearinghouse mechanism at futures exchanges would naturally facili-

tate perpetual contracts. In contrast, if a commercial bank that has taken oneside of a perpetual swap gets out of the swap by a reversing trade with athird party, it is still locked into two contracts forever. The long-term creditrisks of the counterparties is an essential worry for them in these circum-

stances, discouraging the writing of perpetual swaps.

D. Applications of Perpetual Futures

Regional or occupational category labor cost perpetual futures contractsought to fill important hedging needs. Firms may beespecially interested inhedging the present value of their labor costs and workers, on the other sideof futures contracts, interested in hedging the present value of their wages. Afutures price that represents such a present value of wages in the futureisespecially advantageousin the price discovery that it affords. Firms decidingwhere to locate their operations, and individuals deciding where to move, canuse the price discovery afforded by the present value futures price.

Regional commercial real estate perpetual futures contracts may also fillimportant hedging needs. Ordinary futures contracts cash settled on commer-cial real estate prices are problematical; cash price indices used to settle

futures contracts are unreliable since it is notoriously difficult to infer pricechanges in commercial real estate from observed transaction prices for thereal estate. Transactions of commercial real estate are relatively infrequent.Often the property is altered between sales, other property is transmitted attime of sale, and financial deals (such as seller financing at nonmarket rates)accompany the transfer. For these reasons, published indices of commercialreal estate prices are based on appraisals. Unfortunately, appraisals are notunbiased estimates of the price of the property; they are just guesses made bycertain parties, who often have an indirect interest in the property. A hedonicrepeated measures price index might be constructed from transactions datathat takes account of changed characteristics between sales, but doing this

requires that characteristics be measured. It may be more advantageous tocreate a perpetual futures contract based on some measure of rents ofcommercial property. Rents are observed regularly, and so many of them are


paid that it is possible for the constructor of a rental index to be choosy andselect only those rental properties for which the rents observedarerelatively

clean (or to use a hedonic repeated measures index method to correct forunrepresentativeness).

Note that rents on commercial properties tend to be established as parts ofcontracts that may extend for several years or more. This means thatcommercial rents are sluggish and do not respond at all quickly to new

information, all the more reason why any futures contract settled on rents

should be a perpetual futures contract (in contrast to the commercial prop-erty rental index futures that was tried at the London Futures and OptionsExchange in 1991). Some indices of commercial property rents report on rentsof newly negotiated contracts, to try to make their index more forwardlooking; in doing this they lose sample size, and run therisk that the newlynegotiated contracts are somehow systematically different from all contracts.For a perpetual futures contract, even though rents may be set in multiyearcontracts, the rental index could be nothing more than the average rentactually in force on all unchanged properties, or, better, it could be derivedfrom data on all rents from a hedonic repeated measures method like thatdescribed in the preceding section.Farmers may find that it is more in their interest to hedge the price of their

farms than the price of the next crop. But agricultural land prices also poseproblems for constructors of indices, especially as farm sizes contract withtime, so that the component of agricultural land sales that is due to the landitself, and not the house situated on the land, is declining through time.Anyway, farm sales occur less frequently than rental payments, and so there

are more data on rents. Moreover, it might be possible to create a land rentindex froi.. rental or sharecropping contracts in which only the landis rented,and not the house situated onit.Other commodity cash prices might also in some circumsiices be used as

the variable d, in the settlement of a perpetual futures contract. For farmsthat produce single crops, the cash price of the crop may be a reliable

indicator of the rent on the land. The perpetual futures contract may thusbetter serve the hedging needs of owners of such farms, who with today’scontracts have the option only of hedging the next “dividend” rather thantheir farming operation.The consumer price index may be considered more near!» a dividend or

rent than a price of an asset, and so consumerprice index futures might best

be handled as a dividend in the perpetual futures contract. There may belittle uncertainty about the next few values of the consumerprice index, justas there maybelittle uncertainty about the next few dividends to be paid oncorporate stocks, and so a conventional futures market may show little

volatility. A perpetual futures contract might be especially useful to peoplehedging the present value of long-term nominal contracts, such as long-termbonds. And, someone who has purchased a perpetual futures contract basedon any other index might, if the perpetual futures contract uses the nominalrisk-free rate to settle, want to go short in the consumerprice index perpetual


futures contract, so as to convert the other perpetual futures contract from anominal edge into a real hedge.Perpetual futures might even, in some circumstances, be settled on asset

prices rather than dividends or rents. This might be advantageous in caseswhere the cash marketis very illiquid, as with the market for singie familyhomes. If single-family home prices move sluggishly in response to news, thena short horizon futures contract based on an index of selling prices may notreflect the news when it arrives, since the news may not be incorporated intothe cash price before the contract matures. Rolling over conventional futurescontacts on residential real estate may thus not hedge against such news;

subsequent futures prices will already have taken account of such news, so

that it is not hedgeable later.

Ei, Antecedents of Perpetual Futures

Perpetual futures should not be confused with undated futures, such as theundated gold futures traded at the Chinese Gold and Silver Exchange ofHong Kong (see Gehr (1988)). These undated futures are really essentiallyone-day futures, that are automatically rolled over every day uniess the

trader opts to discontinue the contract.’ The daily settlement of the undatedfutures is the change in the spot price minus an “interest” that was deter-mined the previous day in the futures market. Clearly, the interest isequivalent to a one-day futures price minus the spot price. These undatedfutures may be viewed as at the other end of a spectrum from perpetualfutures, the shortest rather than the longest futures.Some commodity swaps have features resembling perpetual claims, but

they are not perpetual; these tend to have maturities ranging from one monthto five years. Commodity swaps have been contracts to provide, in effect, afixed flow of a commodity (suchas oil or gasoline) in exchangeforfixed flow ofcash payments; in practice the contracts are usually cash settled each period.If the cash price in the contract is capitalized into a present value by a

riskless interest rate of matching duration, then that transformed price maybe considered a sort of price for the present value of future commodity prices,though only for the duration of the contract.’®The closest we have come to perpetual futures are the index participations

(IPs) traded May to August 1989 at the American Stock Exchange (AMEX)and the Philadelphia Stock Exchange (PHLX). These IPs were attempts tomake stock price indices marketable as a basket, following recommendationsmade in response to the 1987 stock market crash by the Securities and

’ The Toronto Stock Exchange 300 Composite Spot Index contract at the Toronto FuturesExchange is also a one-day futures market, but differs from the Chinese Gold and Silver

Exchange undated gold futures contract in that the contract has to be reestablished every day.

(Trade has virtually disappeared in this market.) .16 Equity swaps, whose use has been growing rapidly in recent years, have actually generally

used the capital gains component as measured using stock price indices, rather than dividend

component of equity returns, and hence these do not closely resemble perpetual futures.


Exchange Commission (SEC).'” Those traded at the AMEX were calledEquity Index Participations and those traded at the PHLX were called CashIndex Participations.’® The short time span that these were traded wasdefined by regulators and the courts: the [Ps began trading when they wonSEC approval and stopped trading when the U.S. Court of Appeals in Chicagorules that they were under the jurisdiction of the Commodity Futures Trad-ing Commission rather than the SEC.The AMEX and PHLX IPs were fundamentally different from other actual

and proposed market basket products (the New York Stock Exchange’sExchange Stock Portfolios, the Toronto Stock Exchange Toronto 35 IP Units,the Dresdner Bank’s Deutscher Aktienindex IPs and the SuperUnits and theStandard and Poor’s Depositary Receipts traded at the AMEX). A person whois long in oneof the latter contracts is a beneficial owner of the shares in theunderlying index. With the AMEX and PHLX IPs, dividends were paid fromthe shorts to the longs rather than by the companies comprising the index; noownership of shares was involved. To the longs, the contract was like aninvestment in the shares comprising the index: the long paid the IP price andreceived amounts proportional te the dividends accruing to the stocks com-

prising the index until he or she closed out the contract; the long needed toput up no margin. The shorts, who were obligated to give to the longs cashpayments proportional to the dividend equivalent on the stock price index,had to put up 150 percent margin and see the margin account credited ordebited as the IP price changed.There was an important difference between the [Ps at the AMEX and

PHLXandthe perpetual futures discussed here. The underlying asset for IPswas very liquid and had a price measured by an index; the exchanges added aprovision to the contract, a cash-out option for the longs, that enforced somecorrespondence between the IP price and the index. The option representedanother way longs could close out their position (other than selling their IP):they could opt to receive the underlying value proportional to the index,rather than the IP price. When a long exercised this option, there was arandom assignment to a short who had to take the other side and pay anamount proportional to the index. This option was created out of a fear thatthe IP price might not track the stock price index; the option would have theeffect of keeping the IP price from falling below the index value.’” In practice,the specialist assigned to the IPs at the AMEX kept the IP price very close tothe index value when these IPs were traded. There was thus no important

price discovery afforded by the AMEX and PHLX IP markets; in contrast,

See The October 1987 Market Break, U.S. Securities and Exchange Commission (1988), pp.

3-18.

8 See Kupiec (1990) for a description of these instruments.

9 Shorts did not have a cash-out option; the exchanges wanted the IPs to be viewed asinvestment media by longs, and did not want them to be bothered with cash-outs initiated byshorts.


such price discovery might even be described as the raison d’etre for the

perpetual futures markets proposed here.

F. Starting Perpetual Futures

The IPs at the AMEX and the PHLX had an advantage over proposedperpetual futures contracts in that there the cash price was very well known.Getting a perpetual futures contract launched where the cash price is notwell observed might require somedifferent methods than with conventionalfutures markets, since there will be great uncertainty about the initialmarket price. Procedures analogous to those used in the underwriting of

initial public offerings, which help the market deal with the inherent uncer-tainty at the offering date, may be transferred and modified here: publicityprograms by underwriters andelicitations of initial interest by underwritersto help them gauge offering price. Participants in the perpetual market mayalso require conditions limiting amounts of cash settlement in the event of amarket failure, since, given that these are in perpetual contracts, partici-

pants may have heightened concerns that the market may becomeilliquid or

be closed down.

Ill. Summary and Conclusion

Both proposals made here are intended as solutions to measurementproblems, and so their application must hinge on the nature of the measure-ment problems encountered.

In deciding how and whether to apply the first proposal, that of hedonicrepeated measures indices, one must assess the amount of repeated measuresdata, the measurability of quality characteristics, and the nature of themeasurement problems afforded by the time variation in prices of qualitycharacteristics. There is no point in using the repeated measures methodunless there are a large number of repeated measures; ultimately, if mea-

sures are infrequent, this means a long enough historical sample period.There is no point in trying take account of a quality characteristic unless thischaracteristic is both measurable and has a changing impact on asset price or

rent through time.In deciding whether to apply the second proposal, deciding whether to

establish a market in perpetual futures rather than conventional futures, onemust assess which is easier to measure: the true asset price or the dividendon the asset. If the former, then conventional futures contracts should suffice;if the latter, perpetual futures. Failure to measure true asset price may occurfor any of a number of reasons: unrepresentativeness of the assets sold,infrequency of sales, difficulty in measuring what is sold (e.g., financing or

service deals with the sale), or illiquidity and inefficiency of the cash market.On the other hand, failure to measure dividend accurately can also occur, as.

when only an unrepresentative sample of the asset passes through the rental


market, or when tax considerations pertaining to the asset are not reflectedin rents. Unrepresentativeness of the sample of assets passing through thesale or rental markets could be dealt with using the hedonic repeatedmeasures index number construction method, but only if the characteristicsof the assets are available as hedonic variables.

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